is a freely and cyclically reduced word.
then For that reason X in (1) is usually assumed to be finite where one-relator groups are discussed, in which case (1) can be rewritten more explicitly as where
Recall that r is a freely and cyclically reduced word in F(X).
is called a Magnus subgroup of G. A famous 1930 theorem of Wilhelm Magnus,[1] known as Freiheitssatz, states that in this situation H is freely generated by
Here we assume that a one-relator group G is given by presentation (2) with a finite generating set
and a nontrivial freely and cyclically reduced defining relation
is not a proper power (and thus s is also freely and cyclically reduced).
Then the following hold: Starting with the work of Magnus in the 1930s, most general results about one-relator groups are proved by induction on the length |r| of the defining relator r. The presentation below follows Section 6 of Chapter II of Lyndon and Schupp[27] and Section 4.4 of Magnus, Karrass and Solitar[28] for Magnus' original approach and Section 5 of Chapter IV of Lyndon and Schupp[29] for the Moldavansky's HNN-extension version of that approach.
[30] Let G be a one-relator group given by presentation (1) with a finite generating set X.
(since otherwise G is cyclic and whatever statement is being proved about G is usually obvious).
The main case to consider when some generator, say t, from X occurs in r with exponent sum 0 on t. Say
in G is Magnus' original approach exploited the fact that N is actually an iterated amalgamated product of the groups
, amalgamated along suitably chosen Magnus free subgroups.
His proof of Freiheitssatz and of the solution of the word problem for one-relator groups was based on this approach.
Later Moldavansky simplified the framework and noted that in this case G itself is an HNN-extension of L with associated subgroups being Magnus free subgroups of L. If for every generator from
and the inductive step is usually easy to handle in this case.
Moldavansky observed that in this situation is an HNN-extension of L. This fact often allows proving something about G using the inductive hypothesis about the one-relator group L via the use of normal form methods and structural algebraic properties for the HNN-extension G. The general case, both in Magnus' original setting and in Moldavansky's simplification of it, requires treating the situation where no generator from X occurs with exponent sum 0 in r. Suppose that distinct letters
The one-relator group G' can then be treated using Moldavansky's approach.
splits as an HNN-extension of a one-relator group L, the defining relator
of L still turns out to be shorter than r, allowing for inductive arguments to proceed.
Magnus' original approach used a similar version of an embedding trick for dealing with this case.
It turns out that many two-generator one-relator groups split as semidirect products
This fact was observed by Ken Brown when analyzing the BNS-invariant of one-relator groups using the Magnus-Moldavansky method.
is a freely and cyclically reduced word.
be the minimum and the maximum subscripts of the generators occurring in
is an epimorphism with a finitely generated kernel, then G splits as
Later Dunfield and Thurston proved[32] that if a one-relator two-generator group
is chosen "at random" (that is, a cyclically reduced word r of length n in
is chosen uniformly at random) then the probability
with a finitely generated kernel exists satisfies for all sufficiently large n. Moreover, their experimental data indicates that the limiting value for